You can still compute the estimator with sparse inputs because the regularization term ensures the denominator is full rank. If the zeros are standing in for missing values, however, your estimates will be biased.

As for your second question, W* computed from only columns 2 and 4 will only yield the same values as W in the unrestricted model if the columns of X are orthogonal. Could you work with an orthogonal transform (e.g. PCA projection) of the X matrix?